JOURNAL OF CHEMICAL PHYSICS
VOLUME 115, NUMBER 11
15 SEPTEMBER 2001
Dissipative particle dynamics for interacting systems
I. Pagonabarragaa) and D. Frenkel
FOM-Institute for Atomic and Molecular Physics (AMOLF), Kruislaan 407, 1098 SJ Amsterdam,
The Netherlands
共Received 13 December 2000; accepted 3 July 2001兲
We introduce a dissipative particle dynamics scheme for the dynamics of nonideal fluids. Given a
free-energy density that determines the thermodynamics of the system, we derive consistent
conservative forces. The use of these effective, density dependent forces reduces the local structure
as compared to previously proposed models. This is an important feature in mesoscopic modeling,
since it ensures a realistic length and time scale separation in coarse-grained models. We consider
in detail the behavior of a van der Waals fluid and a binary mixture with a miscibility gap. We
discuss the physical implications of having a single length scale characterizing the interaction range,
in particular for the interfacial properties. © 2001 American Institute of Physics.
关DOI: 10.1063/1.1396848兴
I. INTRODUCTION
can be derived from a Hamiltonian, DPD includes dissipative
and random forces. These mimic the effect of viscous damping between fluid elements and the thermal noise of the fluid
elements, respectively. Flekko” y and Coveney7 have shown
that, in principle, a particular DPD-like model can be derived
from an atomistic description. However, no such derivations
exist for the commonly used DPD models. Nonetheless, even
without such a link to the underlying microscopics, it has
been shown that thermal equilibrium can be ensured by an
appropriate choice of the ratio between dissipative and random forces.8 The hydrodynamic behavior of the DPD model
has been explored in some detail,9–12 although the link between the mesoscopic and the macroscopic description is not
completely understood.
In conventional DPD, all interparticle forces have the
same finite interaction range r c . Their amplitudes decay according to a weight function w(r i j ) that has been made to
vanish at r c in order to avoid spurious jumps at the cutoff
distance. In this paper, we employ a more general description
of the conservative interactions. In the existing literature, the
conservative forces have usually been assumed to depend
explicitly on the distance between a pair of particles. For the
sake of computational convenience, the conservative forces
between DPD particles are smooth and monotonic functions
of the distance—in fact, the smoothness of the forces is one
of the advantages of DPD. When the forces depend linearly
on the interparticle separation, the equation of state 共EOS兲 of
the DPD fluid is approximately quadratic in the density and
exhibits no fluid-fluid phase transition. Even though the
forces between DPD particles are smooth, they still induce
structure in the fluid 共reminiscent of atomic behavior兲 on a
length scale of order r c . In this respect, the conventional
DPD scheme is similar to other mesoscopic models for nonideal fluids but differs from the—computationally more
demanding—scheme of Flekko” y and Coveney that was previously mentioned.7
Our aim in this paper is to arrive at a formulation of
DPD that allows for a description of the behavior of nonideal
fluids and fluid-mixtures. To this end, we look for a model in
There is a strong incentive to develop ‘‘mesoscopic’’ numerical techniques to model the dynamics of fluids with different characteristic length scales. Mesoscopic simulations
make it possible to analyze processes that take place on
length and time scales that are out of reach for purely atomistic simulations such as Molecular Dynamics 共MD兲. In MD,
one retains the full atomic details in the description of the
system, but at the expense of restricting the studies to short
times. In contrast, models that describe the system at mesoscopic scales, employ a certain degree of coarse graining,
which allows one to analyze longer times. However, care
should be taken that the loss of ‘‘atomic’’ information associated with the coarse-graining process does not lead to unrealistic features on larger length and time scales. In particular, the coarse-grained models should provide an adequate
description of the equilibrium properties of the system. Some
of the mesoscopic models that have been proposed previously in the literature were derived in a systematic way from
underlying microscopic models, as is the case with the
lattice-Boltzmann method,1 which can be viewed as a preaveraged lattice gas model.2 Coming from the opposite side,
smoothed particle dynamics was introduced as a Lagrangian
discretization of the macroscopic equations of fluid motion.3
A different strategy to simulate structured fluids is to assume
that the solvent is passive, and that the suspended objects
have a diffusive dynamics with diffusion coefficients that are
known a priori. This has led to the development of
Brownian4 and Stokesian dynamics.5
In the early nineties, Dissipative Particle Dynamics
共DPD兲 was introduced as a novel way to simulate fluids at a
mesoscopic scale.6 In DPD, the fluid is represented by a large
number 共N兲 of point particles that have a pairwise additive
interaction. The interparticle forces are the sum of three contributions. In addition to the usual conservative forces that
a兲
Current address: Department de Fı́sica Fonamental, Universitat de Barcelona, Av. Diagonal 647, 08028-Barcelona, Spain
0021-9606/2001/115(11)/5015/12/$18.00
5015
© 2001 American Institute of Physics
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J. Chem. Phys., Vol. 115, No. 11, 15 September 2001
I. Pagonabarraga and D. Frenkel
which there is a direct link between the macroscopic equation of state and the effective interparticle forces. As we will
show, as an additional advantage, our approach results in
rather weak structural correlations in the fluid. In the next
section, we describe in detail the model and how conservative forces are derived. We will subsequently elaborate the
general method on three characteristic examples: A nonideal
fluid without a gas-liquid phase transition that has been studied previously with a different choice of conservative forces,
a van der Waals fluid, and a mixture with a miscibility gap.
In Sec. III we look at the interfacial properties of these examples to gain some insight in the physical meaning of the
conservative forces that we introduce, and subsequently analyze their equilibrium behavior and compare them with previous models. We conclude with a discussion of our main
results.
II. MODEL
In DPD one has N point particles of mass 兵 m i 其 that
interact through a sum of pairwise-additive conservative, dissipative and random forces. These particles can be interpreted as fluid elements, and the dissipative forces are introduced to mimic the viscous drag between them. The random
force equilibrates the energy lost through friction between
the particles, enabling the system to reach an equilibrium
state. To be specific, if we call 兵 rk ,pk 其 the set of particle
positions and momenta of the N point particles, their dynamics are controlled by Newton equations of motion
drk
⫽vk ,
dt
共1兲
dpk
⫽
兵 FC共 ri j 兲 ⫹FD共 ri j 兲 ⫹FR共 ri j 兲 其
dt
j⫽i
兺
⫽
兵 FC共 ri j 兲 ⫺ ␥ ␻ D共 ri j 兲 vi j "ei j ei j ⫹ ␴␻ R共 ri j 兲 ei j ␰ i j 其 ,
兺
j⫽i
共2兲
where we have used the notation ri j ⬅ri ⫺r j and vi j ⬅vi
⫺v j . ei j denotes a unit vector in the direction of ri j , and
vi ⫽pi /m i is the velocity of particle i. The dissipative force,
FD(ri j ), depends both on the relative positions and velocities
of the interacting pair of particles and its amplitude is characterized by the parameter ␥. This parameter is related to the
viscosity of the DPD fluid. The third term in Eq. 共2兲, FR(ri j ),
is a random force acting on each pair of DPD particles—␰
stands for a random variable with Gaussian distribution and
unit variance. The random force has an amplitude ␴ and is
also central. Central pair interactions ensure angular momentum conservation 共although the dynamics can be generalized
to account for noncentral forces13兲. The dissipative and random forces are completely specified once the weight functions, ␻ D(r i j ) and ␻ R(r i j ), are specified—these are smooth
and of finite range. Although they can be chosen at will,
Español and Warren showed8 that ␻ D and ␻ R must be related
to ensure that the probability to observe a particular configuration of DPD particles is given by the Boltzmann distribution in equilibrium. Specifically, if they are chosen such that
␻ R⫽ 冑␻ D, then the correct equilibrium distribution is recovered, and the equilibrium temperature of the DPD fluid is
fixed by the ratio of the amplitudes of the dissipative and
random forces, k BT⫽ ␴ 2 /(2 ␥ ). We stress that the DPD equations of motion, Eqs. 共1兲 and 共2兲, cannot be derived from a
Hamiltonian.
Traditionally, and for simplicity, the conservative forces
in DPD have been taken as pairwise-additive and central,
with a weight function related to ␻ D, and with a variable
amplitude that sets the temperature scale in the system. As
long as the force is sufficiently weak that it does not induce
appreciable inhomogeneities in the density around a DPD
particle, it can only lead to an equation of state with a quadratic dependence in the density, irrespective of the precise
choice for the weight function 共see the following兲. One consequence is that phase separation between disordered phases
cannot occur in a pure system; at least a binary mixture of
different kinds of particles is needed.14
We will first consider the general form that the free energy of a DPD system can have, in order to elucidate the
generic shape of consistent conservative forces. In agreement
with the idea that the DPD particles refer to lumps of fluid, it
seems natural to assume that the relevant energy associated
to their configurations is a free energy, rather than a strictly
‘‘mechanical’’ potential energy. We can express quite generically the free energy F of an inhomogeneous system with
density ␳ (r) as
F⫽
冕
dr␳ 共 r兲 f 共 n 兵 r其 兲 ,
共3兲
where f ( ␳ ) is the expression for the local free energy per
particle 共in units of k BT兲, and n( 兵 r其 ) is related to the density
of the system at r. This formulation is reminiscent of the
strategy followed in density functional theory to study the
equilibrium properties of the fluids.15 In fact, the particular
case n( 兵 r其 )⫽ ␳ ( 兵 r其 ) corresponds to the local density approximation in density functional theory, and if n(r) is chosen to be an average of the density over an interval around r,
it can be understood as a weighted density approximation for
the true free energy. We can separate the total free energy,
f ( ␳ )⫽ f id( ␳ )⫹ f ex( ␳ ), as the sum of the ideal f id( ␳ )
⫽log(⌳3 ␳)⫺1 plus the excess contribution, where ⌳ is the
thermal de-Broglie wavelength. Our purpose is to obtain the
equivalent expression for a DPD system, in which we have N
particles distributed in the space. Since the free energy is an
extensive quantity, the total free energy of a DPD system can
be obviously expressed in terms of the free energy per DPD
particle, ␺, as
N
F⫽
兺
i⫽1
N
␺共 ni兲⫽ 兺
i⫽1
⫽
冕
冕
dr␦ 共 r⫺ri 兲 ␺ 共 n 兵 r其 兲
dr␳ 共 r兲 ␺ 共 n 兵 r其 兲 ,
共4兲
where we have introduced the symbol n i to refer to the generalized density defined previously, although now expressed
in terms of the positions of the discrete N DPD particles 共see
the following兲. Comparing Eqs. 共4兲 and 共3兲, we can easily
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J. Chem. Phys., Vol. 115, No. 11, 15 September 2001
Dissipative particle dynamics
identify ␺ ( ␳ )⫽ f ( ␳ ) which obviously implies that we can
decompose ␺ into its ideal and excess contributions.
If the free energy determines the relevant energy for a
given configuration of DPD particles, we can then derive the
force acting on each particle as the variation of such an energy when the corresponding particle is displaced. However,
the motion of the particles themselves, due to the action of
the dissipative and random forces, already accounts for the
ideal contribution to the free energy of the system, which is
not related to the interactions among the particles. Therefore,
only the excess part of the free energy will be involved in the
effective interactions between the DPD particles. Accordingly, we can write the conservative force acting on particle
i, FCi , as
N
⳵
␺ ex共 n j 兲 .
Fi ⫽⫺
⳵ ri j⫽1
兺
共5兲
We have derived the generic form for the conservative force
acting on a DPD particle as a function of the excess free
energy that characterizes the system, which is, in general, not
a pairwise additive. These forces are analogous to the ones
derived from semiempirical potentials16 in MD, used to effectively model the many-body interactions in condensed
systems. However, we have started from the macroscopic
properties of the system, i.e., its free energy, rather than ensuring microscopic consistency.
We can then fix the equilibrium thermodynamic properties of the system beforehand, and derive a set of conservative forces consistent with the desired equilibrium macroscopic behavior. This procedure is reminiscent of an
approach used in other mesoscopic simulation techniques
that deal with generic nonideal fluids.17
Given that the free energy has been defined as a functional of a certain local density, local variations in such a
density are responsible for the effective forces among the
DPD particles. The particular expression for the forces will
then depend both on the specific form of the free energy and
on the choice of the local density n i . It seems natural to
define the local density of a particle i as its average on the
corresponding interaction range. For simplicity, we weight
this average with the same functions used to define the dissipative and random forces, as introduced in Eq. 共2兲. Therefore, we write
n i⫽
1
关w兴
兺j
w共 ri j 兲,
interaction range. The dependence of the energy of a particular configuration on the particles’ positions enters implicitly
through the weighted densities. For densities of the form
given by Eq. 共6兲, the conservative force acting on particle i
can be rewritten as
N
Fi ⫽⫺
兺
j⫽1
⳵␺共 n j兲
⫽⫺
⳵ ri
w⬘
兺j 共 ␺ i⬘ ⫹ ␺ ⬘j 兲 关 wi 兴j e i j ⬅ 兺j Fi j ,
共7兲
where we have introduced the notation, ␺ i ⬅ ␺ (n i ), and
where the primes denote derivatives with respect to the corresponding variables. Although the free energy of each particle depends on the local density, and leads in general to
many-body effective forces, for the particular local density
introduced in Eq. 共6兲, the forces between DPD particles can
still be written down as additive pairwise forces—a computational advantage.
The fact that the forces depend on the positions of many
particles through their corresponding local weighted densities suggests that, in general, the local structure of the fluid
phase will be smoother than in the case in which forces are
derived from a pair-potential. This is an attractive feature of
the present model; the local structure in a fluid should only
be related to its microscopic structure, and should be
smeared out at mesoscopic, coarse-grained, scales. In this
respect, the density-dependent interactions of these DPD
models enforce an appropriate length scale separation. In the
next sections, we will analyze these properties in detail.
Before considering specific examples, as a consistency
check, we will analyze the predictions for the pressure of a
fluid following the free energy, p th, and the virial, p v, routes.
If we start from the free energy per particle, Eq. 共4兲, the
pressure for a fluid will be
ex
p th⫽⫺ ␳ f ⫹ ␳
⳵ ␺ ex
⳵␳ f
⫽k BT ␳ ⫹ ␳ 2
.
⳵␳
⳵␳
共8兲
On the other hand, since we have derived the force between particles from the free energy, we can also obtain the
pressure of the fluid following the virial route. In this case
the pressure is given
p virial⫽ ␳ k BT⫹
1
2dV
兺i 兺j ri j "Fi j
⫽ ␳ k BT⫹
1
2dV
冕冕
共6兲
where 关A兴 refers to the spatial integral of a given quantity A.
The normalization factor 关w兴 ensures that n i is indeed a density, so that in a homogeneous region, n⫽ ␳ . This is in spirit
similar to the weighted density approximation in density
functional theory.15 The use of a continuous and smooth
weight function that vanishes at the cutoff distance, r c , ensures a smooth sampling of the environment of each particle,
avoiding spurious jumps. There is no a priori reason to
choose w(r) equal to any of the other weight functions, although the particular case of a constant weight function constitutes a pathological limit—in this case the conservative
force will only act when one particle enters or leaves the
5017
drdr⬘ ␳ 共 rr⬘ 兲
⫻ 共 r⫺r⬘ 兲 •F共 r⫺r⬘ 兲 ,
共9兲
where we have approximated the discrete sum over the N
DPD particles by an integral. Introducing the pair correlation
function, g(r), we can rewrite the previous equation as
p
virial
␳2
⫽k BT ␳ ⫹
2d
⫽k BT ␳ ⫺
冕
再
⳵ ␺ ex
⫺2w ⬘ 共 r 兲 e
drg 共 r 兲
r•
⳵␳
关w兴
␳ 2 ⳵ ␺ ex 关 rw ⬘ 兴
.
d ⳵␳ 关 w 兴
冎
共10兲
In the last equality we have assumed that the density is
nearly homogeneous, and that therefore ⳵ ␺ ex/ ⳵␳ is effec-
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5018
J. Chem. Phys., Vol. 115, No. 11, 15 September 2001
I. Pagonabarraga and D. Frenkel
tively a constant. Otherwise, it is not possible to express the
force in terms of the relative coordinates only. If there is no
local structure in the fluid, and w(r c )⫽0, then 关 rw ⬘ 兴 ⫽
⫺d 关 w 兴 , and then Eq. 共10兲 coincides with the prediction for
the ‘‘thermodynamic’’ pressure, Eq. 共8兲 for any weight
function.18 Otherwise, a discrepancy between the two pressures will appear because the averaged density n i is always
centered on the corresponding DPD particle—a conditional
density—and it is therefore related to the g(r). In subsequent sections, we will see in some examples how such local
structure may develop.
Theoretical studies have shown that in the fluid phase of
DPD, in the hydrodynamic limit the usual Navier-Stokes
equation is recovered,9 and that the equilibrium pressure
term is related to the pairwise forces through the usual virial
expression, as we have derived previously. This corresponds
to dynamics which conserves momentum locally 共as in
model-H19兲, instead of being purely relaxational 共as happens
in certain dynamical models that start from density functional theories20兲. By analogy with the usual nonideal DPD
models, in equilibrium we recover a probability distribution
for a given configuration in agreement with Boltzmann fluctuation theorem: The probability of observing a fluctuation is
proportional to the exponential of the deviation of the appropriate thermodynamic potential—the free energy 关as introduced in Eq. 共3兲兴 for DPD models at constant volume, temperature and number of particles.
In the following subsections, we will consider three particular examples, where we will compute explicitly the form
of the conservative forces.
A. Groot and Warren fluid
Let us first derive the expression for the conservative
force that corresponds to the nonideal fluid studied by Groot
and Warren.21 They introduce a conservative force of the
form
Fi j ⫽
再
冉
a 1⫺
0,
冊
rij
e ,
rc ij
r i j ⬍r c
.
共11兲
r i j ⬎r c
For this conservative force, they have shown that the
EOS is p⫽k BT ␳ ⫹ ␣ a ␳ 2 , where by a numerical fit they
found ␣ ⫽0.101⫾0.001. Using the expressions of the previous section, the corresponding pairwise force is
Fi j ⫽
再
2␣a
0,
w ⬘i j
关w兴
ei j ,
r i j ⬍r c
.
共12兲
energy per particle, ␺ ex⫽⫺k BT log兵(1⫺b␳)⫺a␳其. We can recover this EOS in a DPD system with pairwise conservative
forces of the form,
Fi j ⫽
再冉
冊冉
k BTb
k BTb
⫺a ⫹
⫺a
1⫺bn i
1⫺bn j
冊冎
wij
e .
关w兴 ij
共13兲
For reasons that will be discussed below, it is helpful to
generalize slightly the van der Waals fluid allowing for a
contribution cubic in the density. The EOS then becomes p
⫽ ␳ k BT/(1⫺b ␳ )⫺a ␳ 2 ⫺ ␣ 3 ab ␳ 3 . The critical point of this
model corresponds to the parameters
a
T c ⫽ b ␳ c 共 2⫹3 ␣ 3 b ␳ c 兲共 1⫺b ␳ c 兲 2 ,
b
共14兲
2
2
1 ␣ 3 ⫺1⫹ 冑1⫹ 3 ␣ 3 ⫹ ␣ 3
␳ c⫽
,
b
4␣3
␳ c b⬅x c ⫽
⫺1⫹ ␣ 3 ⫹ 冑1⫹ 23 ␣ 3 ⫹ ␣ 23
,
共15兲
T c b/a⬅y c ⫽x c 共 2⫹3 ␣ 3 x c 兲共 1⫺x c 兲 2 .
共16兲
4␣3
The compressibility of the fluid, ␹, in turn, can be written
down as
␹ ⫺1 ⫽
⫽
k BT k BTb 共 2⫺b ␳ 兲
⫹
⫺2a⫺3 ␣ 3 ab ␳
␳
共 1⫺b ␳ 兲 2
yyc
⫺2⫺3 ␣ 3 xx c .
xx c 共 1⫺xx c 兲 2
共17兲
In Fig. 1 we show the behavior of the compressibility for
two different values of the parameter ␣ 3 , for temperature
close to the critical temperature T c . The increase in ␣ 3 reduces ␹ both above and below the critical temperature. As
expected, ␹ becomes negative in a region below T c that is
bounded by a spinodal.
Controlling the compressibility of the fluid is a desirable
feature; a low compressibility helps reducing fluctuations of
the fluid interface, which may be useful in simulations. It
also provides a way of modifying properties of the fluid,
such as the speed of sound. Moreover, it gives an additional
parameter to select the surface tension which, as we will
explain, may even change sign in this DPD-van der Waals
fluid. Finally, it proves useful to reduce the amplitude of the
density fluctuations to compare with mean field theoretical
predictions, as the ones developed in the next section.
r i j ⬎r c
It corresponds to an excess free energy per particle ␺ ex
⫽ ␣ a ␳ , which is linear in the density. As stated in the introduction, an interaction with a smooth, monotonic dependence in position does not induce a fluid-fluid phase separation.
B. van der Waals fluid
The van der Waals fluid is the classic example of a fluid
with a liquid-gas phase transition. It is characterized by the
equation of state p⫽ ␳ k BT/(1⫺b ␳ )⫺a ␳ 2 共and excess free
C. Binary mixture
A binary mixture composed of particles of two species,14
A and B, has also been considered by Groot and Warren.21 In
this system, it is possible to induce demixing with usual pairwise forces by modifying the relative repulsions between the
A – A, B – B, and B – A pairs. Nevertheless, even in this case,
a model in which the forces depend on local densities can be
useful since if they induce less local structure, a relevant
feature at a fluid-fluid interface.
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J. Chem. Phys., Vol. 115, No. 11, 15 September 2001
Dissipative particle dynamics
Fex⫽
冕
⫽
5019
dr兵 2␭ ␳ a 共 r兲 ␳ b 共 r兲 ⫹␭ A ␳ a 共 r兲 2 ⫹␭ B ␳ B 共 r兲 2 其
冋兺
i苸A
共 ␭n B i ⫹␭ A n A i 兲 ⫹
兺
i苸B
册
共 ␭n A i ⫹␭ B n B i 兲 ,
共20兲
where the two sums run over particles of type A and B,
respectively. The corresponding conservative force acting on
particle i can be written down as
Fj ⫽
冋兺 再
i苸A
␭
兺
k苸B
⫹␭ A
兺
k苸A
冎 兺再
⫹
i苸B
␭
兺
k苸A
⫹␭ B
兺
k苸B
⫻关 w ⬘ik eik 共 ␦ i j ⫺ ␦ k j 兲兴 .
冎册
共21兲
Although in this case with two averaged local densities the
conservative forces do not have the form of Eq. 共7兲, they can
still be expressed as pairwise additive forces,
Fi j ⫽
再
⫺2␭ A,B w ⬘i j ei j ,
i j same type
.
i j different type
⫺2␭w ⬘i j ei j ,
共22兲
This fluid will be miscible at high temperatures, and below a
critical temperature T c a miscibility gap will develop. In
terms of the parameters of the free energy, Eq. 共20兲, for a
symmetric mixture T c is
k BT c ⫽ ␳ 共 ␭⫺␭ A 兲 ,
冏
␳A
1
⬅c c ⫽ .
␳ A⫹ ␳ B c
2
共23兲
III. INTERFACIAL BEHAVIOR
FIG. 1. Compressibilities of the van der Waals fluid around the critical
point, for two different values of the parameter ␣ 3 . 共a兲 Curves at T/T c
⫽1.1; 共b兲 Curves at T/T c ⫽0.8.
If the system consists of N A particles of type A and N B
particles of type B, then there are two relevant local density
fields, n A and n B , that are the straightforward generalizations of Eq. 共6兲,
n Ai⫽
兺
j苸A
w共 ri j兲
关w兴
共18兲
In this section we develop a mean field theory for the
interfacial properties for a nonideal DPD fluid that gives
some insight in the meaning of the conservative forces for
these DPD models. For definiteness, we concentrate on the
derivation of the surface tension, ˜␥ .
Since we are interested in the interfacial properties, we
focus on the excess free energy, and will not write down the
ideal gas contribution, which is local in the density and does
not contribute to the interfacial properties. We start from the
continuum limit of the appropriate free energy, and make an
expansion in gradients. Therefore, we disregard correlations
in the positions between the particles, hence the mean field
character of the predictions of the present section.
A. van der Waals fluid
For a van der Waals fluid we can express the continuum
free energy of the fluid, that corresponds to the conservative
forces introduced in Eq. 共13兲, as
Fex⫽
n Bi⫽
兺
j苸B
w共 ri j兲
,
关w兴
共19兲
n A i and n B i represent the concentration of A and B particles
around particle i, respectively. Whenever it is appropriate,
we will denote by ␳ A and ␳ B the continuum limit of the
discrete densities n A and n B , respectively.
The simplest free energy that leads to a miscibility gap
has an excess free energy of the form
冉
冕
dr␳ 共 r兲 ⫺k BT log共 1⫺bn 共 r兲兲 ⫺an 共 r兲
⫺
␣3
abn 共 r兲 2 ,
2
冊
共24兲
where n(r) is the continuum limit of Eq. 共6兲, namely,
n 共 r兲 ⫽
1
关w兴
冕
dr⬘ w 共 兩 r⫺r⬘ 兩 兲 ␳ 共 r⬘ 兲 .
共25兲
In Eq. 共24兲, the density ␳ (r) means the mean density at point
r. This is different from the density appearing in Sec. II,
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5020
J. Chem. Phys., Vol. 115, No. 11, 15 September 2001
I. Pagonabarraga and D. Frenkel
where it referred to the instantaneous value of the density for
a particular configuration. Due to this density preaveraging,
the results of the present section constitute a mean field approximation.
For a smooth planar interface, we can expand the density
in Eq. 共25兲 to second order in the gradients,15
␳ 共 r⫺z兲 ⫽ ␳ 共 r兲 ⫺z•ⵜ ␳ 共 r兲 ⫹ 21 zz:ⵜⵜ ␳ 共 r兲 .
共26兲
Inserting this expression in Eq. 共25兲, and using the fact that
the weight function is radially symmetric we get
n 共 r兲 ⫽ ␳ 共 r兲 ⫹
关 z 2w 兴 2
ⵜ ␳ 共 r兲 .
2d 关 w 兴
共27兲
With this expression, Eq. 共24兲 can be written down as
Fex⫽
冕
⫺a ␳ 共 r兲 ⫺
⫺
冉
再
dr␳ 共 r兲 ⫺k BT ln 1⫺b ␳ 共 r兲 ⫺
冉
b 关 z 2w 兴 2
ⵜ ␳ 共 r兲
2d 关 w 兴
a 关 z 2w 兴 2
ⵜ ␳ 共 r兲
2d 关 w 兴
␣ 3 ab
关 z 2w 兴
␳ 共 r兲 2 ⫹
␳ 共 r兲 ⵜ 2 ␳ 共 r兲
2
d关w兴
冊冎
冊
共28兲
,
where terms containing derivatives higher than second order
have been neglected. Collecting terms in powers of the density gradients, making use of the integration by parts we can
rewrite Eq. 共28兲 in the usual form
Fex⫽
冉
冕
dr␳ 共 r兲 ⫺k BT ln共 1⫺b ␳ 共 r兲兲 ⫺a ␳ 共 r兲
⫺
k BTb
␣3
关 z 2w 兴
⫺
⫹a
ab ␳ 共 r兲 2 ⫹
2
2d 关 w 兴
共 1⫺b ␳ 兵 r其 兲 2
冊
冊
冉
⫹2 ␣ 3 ab ␳ 共 r兲 兩 ⵜ ␳ r兩 2 .
共29兲
The first term in brackets gives the local contribution to the
excess free energy. When the ideal contribution is added, it
gives us the free energy for a homogeneous van der Waals
fluid. The second term in brackets is the energy penalty to
generate gradients in the system. It is this term that contains,
to lowest order, the interfacial energy of the fluid. In particular, we can obtain from it an expression for the surface tension. If we assume that the profile is a hyperbolic tangent,
and we estimate its width from the asymptotic bulk coexisting densities,22 we arrive at
˜␥ ⫽
␳ l⫺ ␳ q
2
冑
冉
冊
d2 f
k BTb
关 z 2w 兴
⫺
⫹a⫹2
␣
ab
␳
,
3
m
d关w兴
d␳2
共 1⫺b ␳ m 兲 2
共30兲
where
d 2 f /d ␳ 2 ⫽1/␳ ⫺2a⫹k BT(2⫺b ␳ )/(1⫺b ␳ ) 2
⫺3 ␣ 3 ab ␳ is the second derivative of the homogeneous free
energy with respect to the density evaluated at one of the
coexisting phases. We have assumed for simplicity that the
density difference between the two phases is small, so that
we can approximate the density across the interface by its
mean value, ␳ m .
If we look at the structure of both the expansion of the
free energy and the surface tension, we can recognize a
qualitative difference with respect to the corresponding expressions for the standard van der Waals fluid. In the latter,
the interfacial tension is a function only of the parameter a
characterizing the long range attraction between the particles,
whereas now it depends on all the parameters, a, b and ␣ 3 .
This qualitative difference can already be traced back to the
coefficient of the gradient square term in free energy expansion, Eq. 共29兲—for the standard van der Waals fluid the gradient energy cost is only related to a. As a result, in this DPD
van der Waals fluid there are different contributions to the
gradient energy term with different signs. Therefore, depending on their relative strength, it is possible either to favor or
penalize the appearance of density gradients in the fluid;
hence, the sign of the interfacial tension may change.
In an atomic fluid, the repulsion parameter, b, in the van
der Waals EOS arises from the hard core repulsion, while the
attraction parameter, a, comes from a long range weak attraction. Therefore, they appear in different length scales,
and accordingly, only the parameter a—related to the longrange structure—is responsible for the behavior of the interfacial tension. On the contrary, for a DPD fluid there is no
excluded volume interaction, and all interactions between the
particles take place at the same length scale, r c . Then, the
relative strength of the different contributions will determine
their overall net effect. It is known that a microscopic model
in which both attractions and repulsions are long ranged
leads to a van der Waals equation of state in which the interfacial behavior can either favor or penalize the presence of
interfaces.23 The van der Waals fluid introduced in this paper
shares these same properties. Even if we can ensure a van der
Waals EOS for a fluid, a careful tuning of the parameters, a,
b and ␣ 3 may lead to a van der Waals model for lamellar
fluid, when even interfaces are favored. Although unrealistic
for atomic fluids, this behavior is relevant, e.g., for nanoparticles, for which repulsive and attractive interactions act on
similar length scales.24
Therefore, depending on the kind of fluid that needs to
be modeled at mesoscopic scales, the parameters in the free
energy should be chosen appropriately. For example, in order
to get a positive surface tension, the densities of the fluid
phases is restricted because one must ensure that both the
pressure and the surface tension are positive. In Fig. 2 we
display the curves where the pressure and the surface tension
vanish for two different values of ␣ 3 . The area defined in
between the corresponding set of curves defines the region of
phase space where the fluid is mechanically stable with a
positive surface tension. Remember that the values of a and
b set the critical values ␳ c and T c . The allowed regions do
not change very much as the parameter ␣ 3 is modified.
If we make b⫽0, this model reduces to that of Groot and
Warren. In this case, ˜␥ becomes negative 共remember that a is
negative now兲. As we have mentioned in Sec. II A, there is
no fluid-fluid phase separation in this model; therefore this
negative value of the surface tension does not lead to a proliferation of interfaces. However, the negative value of ˜␥
implies that the structure factor will have a minimum at a
finite wave vector. We can define a characteristic length, l̃ 0 ,
on which local structure in the fluid will develop. If we ex-
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J. Chem. Phys., Vol. 115, No. 11, 15 September 2001
Dissipative particle dynamics
5021
which does not depend on the amplitude a; only on the shape
of the weight function w. Except for rapidly decaying weight
functions, this length is of order of the interaction range r c .
This fact is consistent with the local structure observed in the
radial distribution functions for this model 共see Sec. IV A兲.
We have also verified numerically the presence of a minimum in the structure factor S(k).
B. Binary mixture
We can also compute the interfacial tension for a binary
mixture following the procedure of the previous subsection.
The excess free energy in the continuum limit is now
Fex⫽␭
FIG. 2. Curves where the pressure and the surface tension vanish for two
different values of ␣ 3 for a van der Waals fluid. Above the solid curve the
pressure is positive, and below the long dashed curves the surface tension is
positive. The region contained in between the corresponding pair of curves
corresponds to the portion of phase space where the fluid will be mechanically stable, with a positive surface tension. Above the long dashed curves
the surface tension is negative. Two different values of ␣ 3 are considered:
␣ 3 ⫽0 and ␣ 3 ⫽5.
pand the free energy Eq. 共24兲 to next order in gradients, we
can estimate this length to be
l̃ 0 ⬃2 ␲ r c
Fex⫽
冕
冑
关 wr 4 兴
,
12关 wr 2 兴
共31兲
dr2␭ ␳ A 共 r兲 ␳ B 共 r兲 ⫹␭ A ␳ A 共 r兲 2 ⫹␭ B ␳ B 共 r兲 2 ⫹
冕
dr关 ␳ A 共 r兲 n B 共 r兲 ⫹ ␳ B 共 r兲 n A 共 r兲兴
⫹␭ A
冕
dr␳ A 共 r兲 n A 共 r兲 ⫹␭ B
冕
dr␳ B 共 r兲 n B 共 r兲 . 共32兲
It is useful to introduce the total density ␳ and the mole
fraction c of component A as the relevant variables. They are
defined as usual,
␳ A⫽ ␳ c ,
共33兲
␳ B ⫽ ␳ 共 1⫺c 兲 .
共34兲
If we expand the local densities n(r) in the same way as
in Eq. 共27兲, we arrive at the square-gradient approximation
for the free energy,
关 z 2w 兴
兵 ␭ ␳ A ⵜ 2 ␳ B ⫹␭ ␳ B ⵜ 2 ␳ A ⫹␭ A ␳ A ⵜ 2 ␳ A ⫹␭ B ␳ B ⵜ 2 ␳ B 其
2d 关 w 兴
⫽2␭ ␳ 2 c 共 1⫺c 兲 ⫹␭ A ␳ 2 c 2 ⫹␭ B ␳ 2 共 1⫺c 兲 2 ⫹
关 z 2w 兴 2
␳ 兵 ⫺␭cⵜ 2 c⫺␭ 共 1⫺c 兲 ⵜ 2 c⫹␭ A cⵜ 2 c⫺␭ B 共 1⫺c 兲 ⵜ 2 z 其
2d 关 w 兴
⫽2␭ ␳ 2 c 共 1⫺c 兲 ⫹␭ A ␳ 2 c 2 ⫹␭ B ␳ 2 共 1⫺c 兲 2 ⫹
关 z 2w 兴 2
␳ 兵 2␭⫺␭ A ⫺␭ B 兲 兩 ⵜ c 兩 2 其 .
2d 关 w 兴
Assuming that ␳ is constant, and for a symmetric mixture
(␭ A ⫽␭ B ), we get
Fex⫽ ␳ 2
冕 再
dr 2 共 ␭⫺␭ A 兲 c 共 1⫺c 兲
冎
关 z 2w 兴
⫹2
共 ␭⫺␭ A 兲 兩 ⵜc 兩 2 .
2d 关 w 兴
共36兲
Again, the interfacial tension can have either a positive or
negative sign, depending on the relative magnitudes of the ␭
parameters. If ␭ A ⫽␭ B ⫽0, and only the repulsion between
the particles belonging to different species is kept, then the
surface tension has the same sign as ␭, as expected.
The interfacial width ␰ can be obtained taking into account that the concentration profile converges exponentially
共35兲
to its bulk value. This gives us ␰ 2 ⫽4k/F ⬙ , where ␬/2 is the
amplitude of the 兩 ⵜc 兩 2 in the gradient expansion of the free
energy, and F ⬙ is the second order derivative of the free
energy with respect to the concentration evaluated at its bulk
coexisting value. In the symmetric case, we get
␰ 2⫽
冉
T
关 z 2␻ 兴
⫺1⫹
␻
4T
c
关 兴
c ⬁ 共 1⫺c ⬁ 兲
冊
⫺1
,
共37兲
where c ⬁ is the value of the concentration in the bulk phase.
The surface tension, ␥, can be obtained integrating the difference between the free energy profile and its bulk value. In
the small gradient limit, it reduces to
␥⫽
冕
⬁
⫺⬁
␳
关 z 2␻ 兴
T 兩 ⵜc 兩 2 .
2d 关 ␻ 兴 c
共38兲
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5022
J. Chem. Phys., Vol. 115, No. 11, 15 September 2001
I. Pagonabarraga and D. Frenkel
FIG. 3. Pressure as a function of the density for a Groot-Warren fluid, using
both the previously proposed pairwise force, Eq. 共11兲, and for the force of
the present form, Eq. 共12兲. In the second case we compare the behavior for
a linear and a quadratic weight function. a⫽25, ␣ ⫽0.101. L⫽6r c , k BT
⫽1, ␥ ⫽1 共see head of Sec. IV for units兲.
If we assume that the concentration profile is a tanh, we get
the estimate
␥⫽
2 关 z 2␻ 兴 ␳ T c
共 c ⬁ ⫺1/2兲 2 .
3关␻兴 ␰
共39兲
Close to the critical point, we recover the expected limiting
behavior for the interfacial properties,22
␥ ⫽2 ␳ k BT c
␰⫽
冑
Tc
T
冑
冉 冊
T
2 关 z 2␻ 兴
1⫺
3d 关 ␻ 兴
Tc
关 z 2w 兴
.
4d 关 w 兴共 1⫺T/T c 兲
3
,
共40兲
共41兲
IV. EQUILIBRIUM PROPERTIES
We will now analyze the equilibrium properties of the
examples of nonideal DPD systems introduced in Sec. II and
will compare with the predictions of previous models performing numerical simulations. We take the interaction range
r c as the unit of length and the mass of the DPD particles m
as the unit of mass. The equations of motion are integrated
self-consistently to avoid spurious drifts in the thermodynamic properties.10
A. Groot and Warren fluid
Before studying a DPD model with fluid-fluid coexistence, we compare the results of our model for a GrootWarren fluid with the original one, based on forces given by
Eq. 共11兲. In this case, both models should coincide and we
analyze it to see the effects of the weight function shape on
the properties of nonideal fluids.
We have performed simulations for a DPD fluid in three
dimensions,
taking
as
parameters
a⫽25
and
␣ ⫽0.101—which corresponds to those used in Ref. 21. In
Fig. 3 we compare the predictions for the EOS given by our
FIG. 4. Radial distribution for a Groot-Warren fluid, using both the previously proposed pairwise force, Eq. 共11兲, and for the force of the present
form, Eq. 共12兲. In the second case, we compare the behavior for a linear and
a quadratic weight function. Same parameters as in Fig. 3. The mean density
is ␳ m ⫽3.
model and by running a DPD simulation with the GrootWarren model.
Groot and Warren used the same weight function for all
pairwise forces. The proposed model for this nonideal fluid
neatly shows that, for the present class of models, a linear
weight function is not suitable to sample the local density of
each DPD particle, because it leads to a pairwise conservative force that exhibits a discontinuity at the edge of the
interaction region, r c . We have analyzed the effect of such a
jump on the thermodynamic and structural properties of this
system. To this end, we have considered both decreasing
linear and quadratic w’s.
In Fig. 3 we compare the EOS obtained from simulations; for a quadratic w, our model coincides with that of
Groot-Warren. However, for a linear w, the agreement survives only at low densities. This DPD model has a transition
to a solid state at high densities, and the results obtained
indicate that the location of such a transition is sensitive to
the shape of the weight function—the characteristic force felt
by each particle depends on the shape of w for a given density. In Figs. 4 –5 we compare the radial distribution functions for our model and that of Groot-Warren, and for different w’s, at increasing values of the density. It is clear that the
shape of w plays an important role in the local structure of
the fluid, and will influence the location of the fluid-solid
transition. In Sec. III A we have noted that for the present
model there exists a characteristic length, l̃ 0 , associated with
density fluctuations and which is of order r c . Only for fairly
narrow weight functions will this length become much
smaller than r c .
At low densities, a linear w generates less local structure,
a pleasant feature for a mesoscopic model. However, as the
density is increased, the local structure develops faster for a
linear weight function, leading sooner to a transition to the
ordered phase. The use of a quadratic weight function leads
to results identical to those of the GW model, while a linear
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J. Chem. Phys., Vol. 115, No. 11, 15 September 2001
Dissipative particle dynamics
5023
FIG. 6. Equation of state for a 2-D van der Waals fluid. The different sets of
data points correspond to different temperatures. b⫽0.016, a⫽1.9b, ␣ 3
⫽5, L⫽7r c , ␥ ⫽1 共see head of Sec. IV for units兲.
FIG. 5. Radial distribution function for a Groot-Warren fluid, using both the
previously proposed pairwise force, Eq. 共11兲, and for the force of the present
form, Eq. 共12兲. In the second case, we compare the behavior for a linear and
a quadratic weight function. Same parameters as in Fig. 3. The mean densities are: 共a兲 ␳ m ⫽8 and 共b兲 ␳ m ⫽14.
force tends to smooth the structure at short distances. The
mean repulsion between particles is larger with a linear w
rather than with a quadratic one. Moreover, it seems plausible to assume that the discontinuity in the force induces a
higher sensitivity to local density fluctuations. These results
show how the modifications of the shape of the weight function can be used to fine-tune details of the behavior of a fluid,
once the EOS has been fixed.
B. van der Waals fluid
Next, we focus on the liquid-gas equilibrium properties
of a two-dimensional van der Waals fluid. Taking a homogeneous system, we can analyze the effect of the density fluctuations on the EOS, and compare it with the predictions
coming from the macroscopically assumed EOS. In Fig. 6,
we show the pressure values obtained in simulations run at
fixed homogeneous density, volume and temperature. In this
case we can recover the characteristic van der Waals loop.
The actual coexistence curve should be derived from it using
the equal area Maxwell’s construction. The agreement with
the expected EOS from the macroscopic free energy is very
good, and only small deviations are observed, due to particle
correlations.
We have also analyzed the density and pressure profiles
when we bring into contact a liquid and a gas in the coexistence region. As mentioned in Sec. II, the compressibility of
the fluid, especially in the coexistence region, is very sensitive to the parameter ␣ 3 that characterizes the amplitude of
the term cubic in the pressure. For ␣ 3 ⫽0 the density profiles
tend to fluctuate substantially. Note that our estimates for the
parameters and ranges of stability are all based on a mean
field description, which may be no longer quantitatively correct under such conditions. Due to this, a series of simulations will be needed for each set of selected parameters
whenever a detailed, quantitative comparison, may be required.
When the parameter ␣ 3 is increased 共we have taken the
value ␣ 3 ⫽5兲, imposing an initial slab of liquid in coexistence with a slab of gas the interface remains stable, and the
density fluctuations in the liquid phase are not too large.
In Fig. 7 we show the temperature, pressure and mean
square displacement of the system during the extension of
the simulation. One can see that the temperature does not
shift, and corresponds to its nominally assigned value. The
pressure exhibits important fluctuations, but if we subtract
the normal and tangential components 共in the figure we only
display the averaged pressure兲, their difference, which is
twice the surface tension, gives a value with a well-defined
positive mean. Also the mean square displacement shows
that particles have had the time to diffuse the interfacial
width, which is roughly proportional to the interaction range,
r c , indicating that the droplet is stabilized.
Fig. 8共a兲 shows the density profiles obtained by starting
with a step density profile in the liquid-gas coexistence region, where the numerical errors are smaller than the fluctuations, as in the rest of the plots. The shape of the drop is
stable and the interfaces fluctuate around their initial location, as could be expected. The density ratio between the two
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5024
J. Chem. Phys., Vol. 115, No. 11, 15 September 2001
I. Pagonabarraga and D. Frenkel
FIG. 7. Thermodynamic values of a DPD fluid with a van der Waals EOS
when a liquid is coexisting with the gas phase, in two dimensions. The
initial condition corresponds to a slab of fluid in the y direction in coexistence with a slab of gas. ˜␥ is the interfacial tension, extracted from the mean
pressures, ˜␥ ⫽(L y /2)( P y y ⫺ P xx ). Also displayed the mean-square displacement in units of the interaction range r c . L y ⫽20, L x ⫽3, k BT⫽0.75, a
⫽1.9* b, b⫽0.0156, ␣ 3 ⫽5. The unit of time is the time needed for a DPD
particle to diffuse r c initially 共see head of Sec. IV for units兲.
fluid phases, ␳ liq / ␳ gas⫽4 makes it reasonable to call the two
phases liquid and gas. The density in the gas phase is 10r ⫺2
c ,
which ensures that in both phases the number of interacting
particles is sufficiently high. By looking at the density profiles, one can also observe that the density fluctuations in the
dense phase are small, as expected on the basis of the small
compressibility of the fluid.
Finally, we have also computed the components of the
pressure tensor across the profiles. For an inhomogeneous
fluid there is no unambiguous way of computing the local
components of the pressure tensor; we follow here the procedure described in Ref. 25 and display them in Fig. 8共b兲.
They follow basically the increase in density, exhibiting
larger fluctuations in the liquid phase. In the bulk phases, the
two components of the pressure tensor have to be equal. This
is clearly shown in Fig. 9, where the differences in the two
components are confined to the interfaces, if we compare the
location of the differences with the density profiles of Fig.
8共a兲. Moreover, the increase in fluctuations in the dense
phase is clearly displayed. The equilibration of the drop can
also be monitored by analyzing the time scale at which the
pressure profile becomes symmetric at both interfaces. Together with the pressure differences, we have also plotted in
thin lines the integral of the pressure difference across the
profile. This quantity is the surface tension, and indeed, the
values we get when the profile is equilibrated agree with the
predictions extracted from the mean pressures, displayed in
Fig. 7. We have also computed the excess free energy profile.
Its integral gives us an alternative 共thermodynamic兲 route to
compute the surface tension. We have verified that the values
of the surface tension obtained integrating the excess free
energy profile coincides with the value presented above,
computed along the virial route.
FIG. 8. Equilibrium 共a兲 density and 共b兲 pressure profiles for a 2-D van der
Waals fluid. The initial profile is a step profile. Same parameters as in Fig. 7
共see head of Sec. IV for units兲.
Another appealing feature of these conservative interactions is that their density dependence induces smooth local
structure. Indeed, if we analyze the radial distribution functions for a homogeneous phase, we can see that the structure
in this case is almost nonexistent. When an interface is
present, it is hard to assess the spurious structure that the
model may induce through the density profile. All we can say
is that the decay of the density is monotonic from one phase
to the other, and therefore, avoids spurious structure close to
the interface. Such a structure would be spurious on the mesoscopic scale modeled by the DPD fluid. In contrast, the
onset of structuring of the liquid-vapor interface on an
atomic scale 共beyond the Fisher-Widom line兲 is a real
effect.26
C. Binary mixture
Finally, we have run simulations for a binary mixture
corresponding to the model described in Sec. II C. As in the
previous subsection, we concentrate on the equilibrium properties of the fluid in the coexistence region. We have simulated a 2-D fluid, starting with an initial step profile in con-
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J. Chem. Phys., Vol. 115, No. 11, 15 September 2001
FIG. 9. Profiles of the difference between the normal and tangential components of the pressure tensor along the system, for the pressure profiles of
Fig. 8共b兲. Same parameters as in Fig. 7 共see head of Sec. IV for units兲.
centration. In Fig. 10 we show the evolution of the
temperature and pressure, which remain essentially constant
through the simulation. We also display the root mean square
displacement of the two species. One can clearly see that
after a short initial period, when the species start to feel the
presence of the interfaces their effective diffusion slows
down. The fact that the mean square displacement is much
larger than the interfacial width, which remains of the order
of the interaction range, r c , ensures that the initial configuration has relaxed to its proper equilibrium shape.
We have computed the concentration profiles as a function of time. In Fig. 11 we show the concentration profiles of
one of the species at an initial and late stage of the relaxation
towards the equilibrium coexistence. As was the case in the
van der Waals fluid, the fluctuations are greater in the concentrated phase. Although the concentration of each species
Dissipative particle dynamics
5025
FIG. 11. Profiles of the relative amount of one of the species across the
system, at two different times. These curves have been multiplied by 2 to
avoid confusion with the thin lines. The latter correspond to the normalized
mean density at the same time 共see head of Sec. IV for units兲.
goes basically to zero in one of the two coexisting phases,
the interface does not broaden and keeps its width within r c .
Despite this large concentration gradient, the mean density
barely changes across the interface. These normalized mean
densities are displayed also in Fig. 11 as thin curves. Although a small dip in the normalized mean density appears at
the interfaces, its value is not large compared with the typical
bulk density fluctuations 共which are due to the compressibility of the fluid兲. Again, this indicates that the use of concentration dependent conservative forces suppresses the appearance of spurious structure at interfaces, while still being able
to drive the phase separation.
We can also test the predictions of Sec. III B for the
interfacial properties on the basis of a binary mixture. To this
end, we have integrated numerically Eq. 共38兲 using the concentration profiles obtained from the simulations, and we
have compared the results with the theoretical prediction, Eq.
共39兲. We display the results in Fig. 12, where we have multiplied the theoretical curve by an overall numerical factor,
since the numerical prefactors in Eq. 共39兲 are approximate.
One can observe that the overall good agreement is lost at
small temperatures, where the interface is very sharp, and
close to the critical point, where fluctuations are expected to
play a relevant role.
V. CONCLUSIONS
FIG. 10. Temperature, pressure and mean-square displacements of the two
species as a function of time, for a binary mixture below its critical temperature, T/T c ⫽0.5, and with ␭⫽1, ␭ A ⫽0.2 共see head of Sec. IV for units兲.
We have presented a new way of implementing conservative forces between DPD particles. Rather than assuming a
force that depends on the interparticle separation, we have
introduced a conservative interaction that depends on the local excess free energy. In this way, it is possible to fix beforehand, at the mean-field level, the desired thermodynamic
properties of the system. However, this procedure neglects
the effect of particle correlations. Whenever an accurate
quantitative comparison is needed, a set of numerical simulations will be required to determine accurately the appropri-
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5026
J. Chem. Phys., Vol. 115, No. 11, 15 September 2001
I. Pagonabarraga and D. Frenkel
probably more common on the mesoscopic than in the microscopic domain. In this respect, the models we have introduced are quite flexible because, for a given bulk thermodynamic behavior 共e.g., a given EOS兲, it is still possible to
modify the parameters to control other physical properties.
For example, the mean interaction strength can be changed
by modifying the way in which the local density is sampled,
or for the van der Waals fluid, it is possible to modify the
compressibility 共and hence the speed of sound兲. As in any
diffuse interface model, the typical interfacial width sets a
minimum length scale in the system. For DPD the natural
scale is r c , unless the parameters are chosen carefully.
ACKNOWLEDGMENTS
FIG. 12. Surface tension for a binary mixture at density ␳ ⫽0.5 with a
critical temperature T c ⫽8 and quicomposed, as a function of the temperature. The squares correspond to the expression derived from the mean-field
free energy in the small gradient limit.
ate phase diagram. We could equally use the free energy to
carry out Monte Carlo simulations to analyze the static properties of fluids; this procedure will suffer from similar drawbacks as a result of the ignored particle correlations.
When the free energy per particle depends on the averaged local density, it is possible to recover central pairwise
additive forces—an important computational feature. The
only assumption we have made is that the system is isothermal, although it should be straight forward to generalize it to
include energy transport, along the lines developed
previously.27
These models can be viewed as a dynamical density
functional theory 共DFT兲 for smooth conservative forces with
local momentum conservation. However, since the DPD particles do not have a local structure, these models can only
describe the dynamics at a mesoscopic level, while the usual
dynamical DFT can account for the dynamics down to the
microscopic scale.
In addition to the freedom in the choice of the free energy, this new type of proposed forces leads to weaker structure at short distances. Hence, we can enforce a proper length
and time scale separation, avoiding the appearance of microscopic features of the system at distances of order r c .
At the mean-field level, and using standard techniques, it
is easy to derive expressions for the interfacial properties.
We have shown that the absence of internal structure of the
DPD particles 共implying that all forces act on the same
length scale兲 leads to qualitatively new behavior not present
in atomic fluids. From the physical point of view, it shows
that, for example, the same thermodynamic system can be
tuned to favor macroscopic or microscopic phase separation.
Although it may seem unrealistic, the competition of attractive and repulsive effective potentials on the same length
scales correspond to certain physical situations, and they are
The work of the FOM Institute is part of the research
program of ‘‘Stichting Fundamenteel Onderzoek der Materie’’ 共FOM兲 and is supported by the Netherlands Organization for Scientific Research 共NWO兲. The authors acknowledge Pep Español for sending us, at the early stages of this
work, a preprint on a similar model for treating conservative
forces in DPD. The authors also acknowledge P. B. Warren
for enlightening and encouraging discussions and J. Yeomans
and S. Y. Trofimov for helpful comments.
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